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Diffstat (limited to 'content/know/concept/sigma-algebra/index.pdc')
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diff --git a/content/know/concept/sigma-algebra/index.pdc b/content/know/concept/sigma-algebra/index.pdc index 690c4cc..1a459ea 100644 --- a/content/know/concept/sigma-algebra/index.pdc +++ b/content/know/concept/sigma-algebra/index.pdc @@ -43,7 +43,7 @@ is a sub-family of a certain $\mathcal{F}$, which is a valid $\sigma$-algebra in its own right. -## Notable examples +## Notable applications A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$, which is defined when $\Omega$ is a metric space, @@ -54,6 +54,8 @@ and all the subsets of $\mathbb{R}$ obtained by countable sequences of unions and intersections of those intervals. The elements of $\mathcal{B}$ are **Borel sets**. +<hr> + Another example of a $\sigma$-algebra is the **information** obtained by observing a [random variable](/know/concept/random-variable/) $X$. Let $\sigma(X)$ be the information generated by observing $X$, @@ -84,6 +86,32 @@ if $Y$ can always be computed from $X$, i.e. there exists a function $f$ such that $Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$. +<hr> + +The concept of information can be extended for +stochastic processes (i.e. time-indexed random variables): +if $\{ X_t : t \ge 0 \}$ is a stochastic process, +its **filtration** $\mathcal{F}_t$ contains all +the information generated by $X_t$ up to the current time $t$: + +$$\begin{aligned} + \mathcal{F}_t + = \sigma(X_s : 0 \le s \le t) +\end{aligned}$$ + +In other words, $\mathcal{F}_t$ is the "accumulated" $\sigma$-algebra +of all information extractable from $X_t$, +and hence grows with time: $\mathcal{F}_s \subset \mathcal{F}_t$ for $s < t$. +Given $\mathcal{F}_t$, all values $X_s$ for $s \le t$ can be computed, +i.e. if you know $\mathcal{F}_t$, then the present and past of $X_t$ can be reconstructed. + +Given some filtration $\mathcal{H}_t$, a stochastic process $X_t$ +is said to be *"$\mathcal{H}_t$-adapted"* +if $X_t$'s own filtration $\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t$, +meaning $\mathcal{H}_t$ contains enough information +to determine the current and past values of $X_t$. +Clearly, $X_t$ is always adapted to its own filtration. + ## References |